Regarding the \(2n-1\)  remaining pieces after folding, we have two options
(in both cases, the number of operations involved is negligible compared to the previous folding operations).
To directly discriminate the point with respect to \(f\) ,
we implement the HLD on these remaining pieces with two additional hidden layers
(as in Figure REF ):
project \(y_{folded}\)  on the \(2n-1\)  hyperplanes (see Theorem REF ), with one layer of width \(2n +1\) ,
and compute the associated Boolean equation with an additional hidden layer.
If needed, we can evaluate \(f(\tilde{y})\)  via \(\mathcal {O}(\log (n))\)  additional hidden layers.
First, compute the \(n-1\)  2-\(\vee \)  via two layers of size \(\mathcal {O}(n)\)  containing several “max ReLU neural networks"
(see e.g. Figure 3 in [1]}). Then, compute the \(n\) -\(\wedge \)  via \(\mathcal {O}(\log (n))\)  layers.
